Your data matches 17 different statistics following compositions of up to 3 maps.
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Mp00252: Permutations restrictionPermutations
St000155: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 0
[2,1] => [1] => 0
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Mp00252: Permutations restrictionPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000021: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 2
[2,3,4,1] => [2,3,1] => [3,2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 2
[3,1,2,4] => [3,1,2] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => [2,3,1] => 1
[3,2,4,1] => [3,2,1] => [2,3,1] => 1
[3,4,1,2] => [3,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => [2,3,1] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 2
[4,3,1,2] => [3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => [2,3,1] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Mp00252: Permutations restrictionPermutations
Mp00240: Permutations weak exceedance partitionSet partitions
St000211: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => {{1}}
=> 0
[2,1] => [1] => {{1}}
=> 0
[1,2,3] => [1,2] => {{1},{2}}
=> 0
[1,3,2] => [1,2] => {{1},{2}}
=> 0
[2,1,3] => [2,1] => {{1,2}}
=> 1
[2,3,1] => [2,1] => {{1,2}}
=> 1
[3,1,2] => [1,2] => {{1},{2}}
=> 0
[3,2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3,4] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,2,4,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2,4] => [1,3,2] => {{1},{2,3}}
=> 1
[1,3,4,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,4,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,4,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3,4] => [2,1,3] => {{1,2},{3}}
=> 1
[2,1,4,3] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1,4] => [2,3,1] => {{1,2,3}}
=> 2
[2,3,4,1] => [2,3,1] => {{1,2,3}}
=> 2
[2,4,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[2,4,3,1] => [2,3,1] => {{1,2,3}}
=> 2
[3,1,2,4] => [3,1,2] => {{1,3},{2}}
=> 1
[3,1,4,2] => [3,1,2] => {{1,3},{2}}
=> 1
[3,2,1,4] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,4,1] => [3,2,1] => {{1,3},{2}}
=> 1
[3,4,1,2] => [3,1,2] => {{1,3},{2}}
=> 1
[3,4,2,1] => [3,2,1] => {{1,3},{2}}
=> 1
[4,1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[4,1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[4,2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[4,2,3,1] => [2,3,1] => {{1,2,3}}
=> 2
[4,3,1,2] => [3,1,2] => {{1,3},{2}}
=> 1
[4,3,2,1] => [3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4,5] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,3,5,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3,5] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,2,4,5,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,2,5,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,5,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4,5] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,2,5,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2,5] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,3,4,5,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,3,5,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,5,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3,5] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,2,5,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,3,2,5] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,5,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,5,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,5,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
St000703: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [3,1,2] => 2
[2,3,4,1] => [2,3,1] => [3,1,2] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [3,1,2] => 2
[3,1,2,4] => [3,1,2] => [2,3,1] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => [2,3,1] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [3,1,2] => 2
[4,3,1,2] => [3,1,2] => [2,3,1] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 1
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
Mp00252: Permutations restrictionPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000325: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[3,2,4,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
Mp00252: Permutations restrictionPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000470: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[3,2,4,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [3,2,1] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
Description
The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.
Matching statistic: St000157
Mp00252: Permutations restrictionPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [[1]]
=> 0
[2,1] => [1] => [1] => [[1]]
=> 0
[1,2,3] => [1,2] => [1,2] => [[1,2]]
=> 0
[1,3,2] => [1,2] => [1,2] => [[1,2]]
=> 0
[2,1,3] => [2,1] => [2,1] => [[1],[2]]
=> 1
[2,3,1] => [2,1] => [2,1] => [[1],[2]]
=> 1
[3,1,2] => [1,2] => [1,2] => [[1,2]]
=> 0
[3,2,1] => [2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2,4] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,4,3,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1,4] => [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[2,3,4,1] => [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,4,3,1] => [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[3,1,2,4] => [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 1
[3,2,4,1] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[4,1,3,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[4,2,3,1] => [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[4,3,1,2] => [3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00252: Permutations restrictionPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00064: Permutations reversePermutations
St000245: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [2,1] => 0
[1,3,2] => [1,2] => [1,2] => [2,1] => 0
[2,1,3] => [2,1] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 0
[3,2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => [1,2,3] => 2
[2,3,4,1] => [2,3,1] => [3,2,1] => [1,2,3] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => [1,2,3] => 2
[3,1,2,4] => [3,1,2] => [3,1,2] => [2,1,3] => 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [1,3,2] => 1
[3,2,4,1] => [3,2,1] => [2,3,1] => [1,3,2] => 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [2,1,3] => 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => [1,2,3] => 2
[4,3,1,2] => [3,1,2] => [3,1,2] => [2,1,3] => 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000662
Mp00252: Permutations restrictionPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
St000662: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [3,2,1] => [3,2,1] => 2
[2,3,4,1] => [2,3,1] => [3,2,1] => [3,2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 2
[3,1,2,4] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[3,2,4,1] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 2
[4,3,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [3,1,2] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 1
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Mp00252: Permutations restrictionPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
Mp00066: Permutations inversePermutations
St000213: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,3,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[2,1,3] => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[3,1,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[3,2,1] => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,2,4,1] => [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[4,1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
[1,3,5,2,4] => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,3] => [3,4,2,1] => [4,3,1,2] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as $$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000702The number of weak deficiencies of a permutation. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001427The number of descents of a signed permutation. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car.